Theorem elfv2ex

Description: If a function value of a function value has a member, then the first argument is a set. (Contributed by AV, 8-Apr-2021)

	Ref	Expression
Assertion	elfv2ex	⊢ ( 𝐴 ∈ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝐶 ) → 𝐵 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( 𝐵 ∈ V → ( 𝐴 ∈ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝐶 ) → 𝐵 ∈ V ) )
2		fv2prc	⊢ ( ¬ 𝐵 ∈ V → ( ( 𝐹 ‘ 𝐵 ) ‘ 𝐶 ) = ∅ )
3	2	eleq2d	⊢ ( ¬ 𝐵 ∈ V → ( 𝐴 ∈ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝐶 ) ↔ 𝐴 ∈ ∅ ) )
4		noel	⊢ ¬ 𝐴 ∈ ∅
5	4	pm2.21i	⊢ ( 𝐴 ∈ ∅ → 𝐵 ∈ V )
6	3 5	syl6bi	⊢ ( ¬ 𝐵 ∈ V → ( 𝐴 ∈ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝐶 ) → 𝐵 ∈ V ) )
7	1 6	pm2.61i	⊢ ( 𝐴 ∈ ( ( 𝐹 ‘ 𝐵 ) ‘ 𝐶 ) → 𝐵 ∈ V )